lemmaroutine

Base height bounded by Faltings height

Bounds base height of

\pi(y)

C(1+\max\{0,h_{\mathrm F}(\mathcal A_y)\})

converts Faltings height control into height of s

Lemma·hei:base-height·§4.4 Comparison of base height and Faltings height

Statement (paper)

For every

y\in Y(\overline\Q)

one has

\hgt(\pi(y))\le C\bigl(1+\max\{0,h_{\mathrm F}(\mathcal A_y)\}\bigr),

with a constant independent of

y

and its field of definition.

In plain English

For every

y\in Y(\overline\Q)

, the height of its image

\pi(y)

on the base is at most

C(1+\max\{0,h_{\mathrm F}(\mathcal A_y)\})

, where

C

is independent of

y

and its field of definition.

Why it's here

This lemma turns control of the Faltings height of the fibers

\mathcal A_y

into control of the height of the base point, which is what the height estimates that use it need. The proof chains three steps: the base height is bounded by Pazuki's theta height via a nonconstant theta-null map on finite covers, the theta-null height agrees with that theta height up to a constant, and the theta height is bounded by the Faltings height.

Classified routine:

Chains standard comparisons of heights: functoriality of Weil heights under the theta-null map on a level cover, and Pazuki's comparison of theta height with Faltings height (Pazuki 2012), in the style of Faltings' height comparisons on moduli of abelian varieties.

Uses

Nothing else in the paper

Used by

thmHeight bounded by radical of contact primesclaimCM fibers have uniformly bounded Faltings height

Inside the proof

claimTheta-null data after finite coversclaimTheta-null height matches Pazuki theta heightclaimTheta-null map nonconstant at some levelclaimBase height bounded by theta heightclaimTheta height bounded by Faltings height

Sources

PaperOpen

Section: §4.4 Comparison of base height and Faltings height
Page: —
Label: hei:base-height
Line: sections/04-height.tex:364

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